Foundation Multi Axis Robustness Axis P Moves D

The dimension selector

In the Recognition Science framework, a recognized object of dimension p lives in a substrate whose codimension is given by the formula 2p + 1. The declaration axis_P_moves_D is the machine-checked statement that this formula is the only axis that can move the substrate dimension away from 3. If p is at least 1 and p is not 1, then the codimension is not 3. The proof is immediate from the definition: 2p + 1 equals 3 exactly when p equals 1.

The framework's library, a machine-checked collection of formal theorems, records this as a robustness result. The dimension route is stable under changes to the recognized-object dimension: only p = 1 yields D = 3, and any other allowed p yields a different codimension. The companion theorem p_one_gives_D3 states that the p = 1 case gives D = 3, and the bundled theorem multi_axis_robustness packages both facts together. A further compatibility theorem shows that when p = 1, the codimension route agrees with the framework's existing forced dimension result.

What the declaration does not claim is just as important. The motivating account names three other axes: C (coefficient ring), I (tracked invariant), and A (acyclicity). These are described as stability claims under perturbation, but they are not theorems in the library. They were once stubbed as trivial true statements, and those stubs were removed in a vacuity quarantine because a statement that cannot fail proves nothing. The library now records them as open obligations, to be formalized only when a richer substrate model exists that can support a statement that might be false.

The practical consequence is narrow but real. Within the framework, the three-dimensionality of the substrate is not a free choice or a coincidence; it is tied to the recognized-object dimension being exactly 1. If that dimension were different, the codimension formula would force a different substrate dimension. The framework's account of why space is three-dimensional therefore rests on this arithmetic selection, and the other candidate axes are explicitly not part of the proof.

/-- If `p ≥ 1` and `p ≠ 1`, the codimension dimension `2p+1` is not `3`. -/
theorem axis_P_moves_D (p : ℕ) (_hp : 1 ≤ p) (hne : p ≠ 1) :
    CodimensionDimension p ≠ 3 := by
  unfold CodimensionDimension
  omega

/-- The `p = 1` codimension case is `D = 3`. -/
theorem p_one_gives_D3 :
    CodimensionDimension 1 = 3 := by
  rfl

/-- Bundled multi-axis robustness theorem (Axis P content).

Only Axis P can move the dimension away from `3`. Axes C, I, and A remain
named stability obligations of the paper; they are not claimed as theorems
here until a non-trivial substrate model exists. -/
theorem multi_axis_robustness :
    (∀ p : ℕ, 1 ≤ p → p ≠ 1 → CodimensionDimension p ≠ 3) ∧
      CodimensionDimension 1 = 3 :=
  ⟨axis_P_moves_D, p_one_gives_D3⟩

/-- Compatibility with the existing dimension forcing result: once `p = 1`,
the codimension route agrees with the existing forced dimension. -/
theorem p_one_route_agrees_with_dimension_forced :
    ∃! D : DimensionForcing.Dimension,
      D = CodimensionDimension 1 ∧ DimensionForcing.RSCompatibleDimension D := by
  refine ⟨3, ?_, ?_⟩
  · constructor
    · rfl
    · exact DimensionForcing.D3_compatible
  · intro D hD
    exact hD.1.trans (by rfl)

What this page does not claim

Axes C, I, and A are not proved as theorems in this module. The declaration does not say that the recognized-object dimension p is itself forced to be 1. The physical bridge from recognition to three-dimensional space is not established by this arithmetic statement alone.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Foundation/MultiAxisRobustness.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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